Ginzburg-Landau vortices, Coulomb Gases, and Renormalized

Energies

Sylvia Serfaty

November 11, 2013

Abstract

This is a review about a series of results on vortices in the Ginzburg-Landau modelof superconductivity on the one hand, and point patterns in Coulomb gases on the otherhand, as well as the connections between the two topics.

keywords: Ginzburg-Landau equations, superconductivity, vortices, Coulomb Gas, one-component plasma, jellium, Renormalized energy.

Most of this paper describes joint work with Etienne Sandier, which has naturally led fromthe study of the Ginzburg-Landau equations of superconductivity – a rather involved systemof PDE – to that of a well-known statistical mechanics system: namely the classical Coulombgas. We will review results in each area and explain the similarity in the mathematics involved.

1 The Ginzburg-Landau model of superconductivity

A type-II superconductor, cooled down below its critical temperature experiences the circu-lation of “superconducting currents” without resistance, and has a particular response in thepresence of an applied magnetic field. Above a certain value of the external field called thefirst critical field, vortices appear. When the field is large enough, the experiments (datingfrom the 60’s) show that they arrange themselves in (often) perfect triangular lattices, cf.http://www.fys.uio.no/super/vortex/ or Fig. 1 below.

Figure 1: Abrikosov lattices

1

These are named Abrikosov lattices after the physicist Abrikosov who had predicted,from the Ginzburg-Landau model, that periodic arrays of vortices should appear [A]. Thesevortices repel each other like Coulomb charges would, while being confined inside the sampleby the applied magnetic field. Their triangular lattice arrangement is the result of these twoopposing effects.

1.1 The model

The Ginzburg-Landau model was introduced on phenomenological grounds by Landau andGinzburg in the 50’s [GL] ; after some nondimensionalizing procedure, in a two-dimensionaldomain Ω it takes the form

(1) GLε(ψ,A) =1

2

∫Ω|(∇− iA)ψ|2 + |∇ ×A− hex|2 +

1

2ε2

(1− |ψ|2

)2This is an idealized situation where the sample is assumed to be a three-dimensional infinitelylong cylinder with cross-section Ω, submitted to an external field parallel to the axis of thecylinder and of intensity hex. Here ψ is a complex-valued function, called “order parameter”and indicating the local state of the sample: |ψ|2 is the density of “Cooper pairs” of super-conducting electrons. With our normalization |ψ| ≤ 1, and where |ψ| ' 1 the material is inthe superconducting phase, while where |ψ| = 0, it is in the normal phase (i.e. behaves likea normal conductor), the two phases being able to coexist in the sample.

The vector field A is the gauge field or vector potential of the magnetic field. The magneticfield in the sample is deduced by h = ∇×A = curlA = ∂1A2 − ∂2A1, it is thus a real-valuedfunction in Ω.

Finally, the parameter ε is a material constant, it is the inverse of the “Ginzburg-Landauparameter” usually denoted κ. It is also the ratio between the “coherence length” usuallydenoted ξ (roughly the vortex-core size) and the “penetration length” of the magnetic fieldusually denoted λ. We are interested in the regime of small ε, corresponding to high-κ (orextreme type-II) superconductors. The limit ε → 0 or κ → ∞ that we will consider is alsocalled the London limit.

This is a U(1)-gauge theory and the functional (as well as all the physically meaningfulquantities) is invariant under the gauge-change

(2)

ψ 7→ ψeiΦ

A 7→ A+∇Φ

where Φ is a smooth enough function.The stationary states of the system are the critical points of GLε, or the solutions of the

Ginzburg-Landau equations :

(GL)

−(∇A)2ψ =

1

ε2ψ(1− |ψ|2) in Ω

−∇⊥h = 〈iψ,∇Aψ〉 in Ω

h = hex on ∂Ω

∇Aψ · ν = 0 on ∂Ω,

where ∇⊥ denotes the operator (−∂2, ∂1), ∇A = ∇−iA, and ν is the outer unit normal to ∂Ω.Here appears the superconducting current, a real valued vector field given by j = 〈iψ,∇Aψ〉

2

where 〈., .〉 denotes the scalar-product in C identified with R2. It may also be written as

i

2

(ψ∇Aψ − ψ∇Aψ

),

where the bar denotes the complex conjugation. For further details on the model, we refer to[GL, DeG, T, FH, SS1].

The Ginzburg-Landau model has led to a large amount of theoretical physics literature— probably most relevant to us is the book by De Gennes [DeG]. However, a precise math-ematical proof of the phase transition at the first critical field, and of the emergence of theAbrikosov lattice as the ground state for the arrangement of the vortices was still missing.

In the 90’s, researchers coming from nonlinear analysis and PDEs became interested in themodel (precursors were Berger, Rubinstein, Schatzman, Chapman, Du, Baumann, Phillips...cf. e.g. [Ch, DGP] for reviews), with the notable contribution of Bethuel-Brezis-Helein[BBH] who introduced systematic tools and asymptotic estimates to study vortices, but inthe simplified Ginzburg-Landau equation not containing the magnetic gauge, and allowingonly for a fixed number of vortices. This was then adapted to the model with gauge but witha different boundary condition by Bethuel and Riviere [BR1, BR2]. It was however not clearthat this approach could work to treat the case of the full magnetic model when the numberof vortices gets unbounded as ε → 0. It is only with the works of Sandier [Sa] and Jerrard[Je] that tools capable of handling this started to be developed. Relying on these tools andexpanding them, in a series of works later revisited in a book [SS1], we analyzed the full modeland obtained the proof of the phase transition, and the computation of the asymptotics of thefirst critical field in the limit ε → 0. We characterized the optimal number and distributionof the vortices and derived in particular a “mean-field regime” limiting distribution for thevortices, which will be described just below. Note that this analysis and the tools developedto understand the vortices have proven useful to study vortices in rotating superfluids likeBose-Einstein condensates (cf. e.g. [CPRY] and references therein), a problem which hasa large similarity with Ginzburg-Landau from the mathematical perspective, and of currentinterest for experiments.

1.2 Critical fields and vortices

What are vortices? A vortex is an object centered at an isolated zero of ψ, around whichthe phase of ψ has a nonzero winding number, called the degree of the vortex. So it isalso a small defect of normal phase in the superconducting phase, surrounded by a loop ofsuperconducting current. When ε is small, it is clear from (1) that any discrepancy between|ψ| and 1 is strongly penalized, and a scaling argument hints that |ψ| is different from 1 only inregions of characteristic size ε. A typical vortex centered at a point x0 behaves like ψ = ρei ϕ

with ρ = f( |x−x0|ε ) where f(0) = 0 and f tends to 1 as r → +∞, i.e. its characteristic core

size is ε, and1

2π

∫∂B(x0,Rε)

∂ϕ

∂τ= d ∈ Z

is an integer, called the degree of the vortex. For example ϕ = dθ where θ is the polar anglecentered at x0 yields a vortex of degree d at x0.

There are three main critical values of hex or critical fields Hc1 , Hc2 , and Hc3 , for whichphase-transitions occur.

3

• For hex < Hc1 there are no vortices and the energy minimizer is the superconductingstate (ψ ≡ 1, A ≡ 0). (This is a true solution if hex = 0, and a solution close to thisone (i.e. with |ψ| ' 1 everywhere) persists if hex is not too large.) It is said that thesuperconductor “expels” the applied magnetic field, this is the “Meissner effect”, andthe corresponding solution is called the Meissner solution.

• For hex = Hc1 , which is of the order of |log ε| as ε→ 0, the first vortice(s) appear.

• For Hc1 < hex < Hc2 the superconductor is in the “mixed phase” i.e. there are vortices,surrounded by superconducting phase where |ψ| ' 1. The higher hex > Hc1 , the morevortices there are. The vortices repel each other so they tend to arrange in thesetriangular Abrikosov lattices in order to minimize their repulsion.

• For hex = Hc2 ∼ 1ε2

, the vortices are so densely packed that they overlap each other,and a second phase transition occurs, after which |ψ| ∼ 0 inside the sample, i.e. allsuperconductivity in the bulk of the sample is lost.

• For Hc2 < hex < Hc3 superconductivity persists only near the boundary, this is calledsurface superconductivity. More details and the mathematical study of this transitionare found in [FH] and references therein.

• For hex > Hc3 = O( 1ε2

) (defined in decreasing fields), the sample is completely in thenormal phase, corresponding to the “normal” solution ψ ≡ 0, h ≡ hex of (GL). See [GP]for a proof.

1.3 Formal correspondence

Given a family of configurations (ψε, Aε) it turns out to be convenient to express the energyin terms of the induced magnetic field hε(x) = ∇ × Aε(x). Taking the curl of the secondrelation in (GL) we obtain

−∆hε + hε = curl 〈iψε,∇Aεψε〉+ hε ' curl∇ϕε

where we approximate |ψε| by 1 and where ϕε denotes the phase of ψε. One formally hasthat curl∇ϕε = 2π

∑i diδai where aii is the collection of zeroes of ψ, i.e. the vortex centers

(really depending on ε), and di ∈ Z are their topological degrees. This is not exact, however itcan be given some rigorous meaning in the asymptotics ε→ 0. We may rewrite this equationin a more correct manner

(3)

−∆hε + hε ' 2π

∑i diδ

(ε)ai in Ω

h = hex on ∂Ω,

where the exact right-hand side in (3) is a sum of quantized charges, or Dirac masses, whichshould be thought of as somehow smeared out at a scale of order ε. This relation is called inthe physics literature the London equation (it is usually written with true Dirac masses, butthis only holds approximately). It indicates how the magnetic field penetrates in the samplethrough the vortices.

4

Some computations (with the help of all the mathematical machinery developed to de-scribe vortices) lead eventually to the conclusion that everything happens as if the Ginzburg-Landau energy GLε of a configuration were equal to

(4) GLε(ψε, Aε) '1

2

∫Ω|∇hε|2 + |hε − hex|2

=1

2

∫∫Ω×Ω

GΩ(x, y)(

2π∑i

diδ(ε)ai − hex

)(x)(

2π∑i

diδ(ε)ai − hex

)(y),

where GΩ is a type of Green (or Yukawa) kernel, solution to

(5)

−∆GΩ +GΩ = δy in Ω

GΩ = 0 on ∂Ω,

and hε solves (3). With this way of writing, and in view of the logarithmic nature of GΩ, onerecognizes essentially a pairwise Coulomb interaction of positive charges in a constant negativebackground (−hex), which is what leads to the analogy with the Coulomb gas described later.There remains to understand for which value of hex vortices become favorable, and with whichdistribution. To really understand that, the effect of the “smearing out” of the Dirac chargesneeds to be more carefully accounted for. Instead of each vortex having any infinite cost in(4) (which would be the case with true Diracs) the real cost of each vortex can be evaluatedas being ∼ πd2

i |log ε| per vortex (roughly the equivalent of the cost generated by a Diracmass smeared out at the scale ε). We may thus evaluate (4) as

(6)GLε(ψε, Aε)

hex2 ' |log ε|

hex

π∑

i d2i

hex+hex

2

2

∫Ω|∇h|2 + |h− 1|2

where h = limε→0hεhex

.Optimizing over the degrees di’s allows to see that the degrees di = 1 are the only favorable

ones. In view of (3), assuming this is true we can then rewrite π∑

i d2i as 1

2

∫Ω | −∆hε + hε|.

Passing to the limit ε → 0, and assuming hex|log ε| → λ as ε → 0, we find that the mean-field

limit energy arising from (4) is

(7)GLε

hex2 'ε→0 EMF

λ (h) =1

2λ

∫Ω| −∆h+ h|+ 1

2

∫Ω|∇h|2 + |h− 1|2

=1

2λ

∫Ω|µ|+ 1

2

∫∫Ω×Ω

GΩ(x, y)d(µ− 1)(x) d(µ− 1)(y)

where h is here related to the limiting “vorticity” (or vortex density) µ := limε→01hex

2π∑

i diδaiby −∆h+ h = µ in Ω with h = 1 on ∂Ω, simply by taking the limit of (3).

In (7) the first contribution to the energy corresponds the total self-interaction of thevortices in (4), while the second one is the cross-interaction of the vortices and the vorticesand the equivalent background charge (which is really the result of the confinement effect ofthe applied magnetic field).

We have the following rigorous statement.

5

Theorem 1 ([SS2], [SS1] Chap. 7). Assume hex ∼ λ|log ε| as ε → 0, where λ > 0 is aconstant independent of ε. If (ψε, Aε) minimizes GLε, then as ε→ 0

2π∑

i diδaihex

→ µ∗ in the weak sense of measures

where µ∗ is the unique minimizer of EMFλ . Moreover

(8) minGLε = hex2(min EMF

λ + o(1)) as ε→ 0.

Minimizing EMFλ leads to a standard variational problem called an “obstacle problem”.

The corresponding optimal distribution of vorticity is uniform of density 1 − 1/(2λ) on asubdomain ωλ of Ω depending only on λ.

An easy analysis of this obstacle problem yields the following (cf. also Fig. 2):

1. ωλ = ∅ (hence µ∗ = 0) if and only if λ < λΩ, where λΩ is given by

(9) λΩ = (2 max |h0 − 1|)−1

for h0 the solution to

(10)

−∆h0 + h0 = 0 in Ωh0 = 1 on ∂Ω.

2. For λ > λΩ, the measure of ωλ is nonzero, so the limiting vortex density µ∗ 6= 0.Moreover, as λ increases (i.e. as hex does), the set ωλ increases. When λ = +∞ (thiscorresponds to the case hex |log ε|), ωλ becomes Ω and µ∗ = 1Ω.

!

"

µ#=1$1/(2!)

µ#=0

%

Figure 2: Optimal density of vortices according to the obstacle problem.

Since Hc1 corresponds to the applied field for which minimizers start to have vortices, thisleads us to expecting that

(11) Hc1 ∼ λΩ|log ε| as ε→ 0

where λΩ depends only on Ω via (9)–(10).

6

In fact this is true, because we were able to show that below this value Hc1 , not only theaverage vortex density µ∗ is 0, but there are really no vortices. To see this, a more refinedasymptotic expansion of GLε than (7) is needed. It suffices instead to note that in the regimewhen a zero or small number of vortices is expected, the solution to (3) is well approximatedby hexh0 where h0 solves (10), and then to split the true hε as hexh0 + h1,ε where h1,ε is aremainder term, and expand the energy GLε in terms of this splitting.

Let us state the result we obtain when looking this way more carefully at the regimehex ∼ λΩ|log ε| and analyzing individual vortices. For simplicity, we assume that the functionh0 achieves a unique minimum at a point p ∈ Ω (this is satisfied for example if Ω is convex)and that its Hessian at that point, Q, is nondegenerate.

Theorem 2 ([Se1, Se2], [SS1] Chap. 12). There exists an increasing sequence of values

Hn = λΩ|log ε|+ (n− 1)λΩ log|log ε|n

+ constant order terms

such that if hex ≤ λΩ|log ε|+O(log |log ε|) and hex ∈ (Hn, Hn+1), then global minimizers of

GLε have exactly n vortices of degree 1, at points aεi → p as ε→ 0, and the aεi :=√

hexn (aεi−p)

converge as ε→ 0 to a minimizer of

(12) wn(x1, · · · , xn) = −∑i 6=j

log |xi − xj |+ n

n∑i=1

Q(xi).

We find here the precise value of Hc1 for which the first vortex appears in the minimizers,and then a sequence of “critical fields” for which a second, a third, etc.. vortices appearin minimizers, together with a characterization of their optimal locations, governed by anexplicit interaction energy wn.

1.4 The next order study

Theorem 1 above proved that above Hc1 , for λ > λΩ, the number of vortices is proportionalto hex and they are uniformly distributed in a subregion of the domain, but it is still far fromexplaining the optimality of the Abrikosov lattice. To (begin to) explain it, one needs to lookat the next order in the energy asymptotics (7), and at the blown-up of (3) at the inverse ofthe intervortex distance scale, which here is simply

√hex. For simplicity, let us reduce to the

case λ = 1 (or hex |log ε|) where the limiting optimal measure is µ∗ = 1Ω and the limitingh ≡ 1.

Once the blow-up by√hex is performed and the limit ε→ 0 is taken, (3) becomes

(13) −∆H + 1 = 2π∑a

δa in R2

where the limiting blown-up points a form an infinite configuration in the plane, and theseare now true Diracs (one may in fact reduce to the case where all degrees are equal to +1,other situations being energetically too costly).

One may recognize here essentially a jellium of infinite size, and E = ∇H the electric fieldgenerated by the points (its rotated vector field j = −E⊥ corresponds to the superconductingcurrent in superconductivity). The jellium model was first introduced by Wigner [Wi2], andit means an infinite set of point charges with identical charges with Coulomb interaction,

7

screened by a uniform neutralizing background, here the density −1. It is also called a one-component plasma.

It then remains first to identify and define a limiting interaction energy for this “jellium,”and second to derive it from GLε. The energy, that will be denoted W , arises as a next ordercorrection term in the expansion of minGLε beyond the order hex

2 term identified by (7)-(8).Isolating efficiently the next order terms in GLε relies on another “splitting” of the energy.

Instead of expanding hε near hexh0 as in the case with few vortices, one should expand aroundhexhλ where hλ is the minimizer of EMF

λ , i.e. the solution to the obstacle problem above.Splitting hε = hexhλ + h1,ε where h1,ε is seen as a remainder, turns out to exactly isolate theleading order contribution hex

2 min EMFλ from an explicit lower order term.

1.4.1 The renormalized energy: definition and properties

We next turn to discussing the effective interaction energy between the blown-up points. Aswe said, it should be a total Coulomb interaction between the points (seen as discrete positivepoint charges) and the fixed constant negative background “charge”. Of course defining thetotal Coulomb interaction of such a system is delicate because several difficulties arise: first,the infinite number of charges and the lack of local charge neutrality, which lead us to definingthe energy as a thermodynamic limit; second the need to remove the infinite self-interactioncreated by each point charge, now that we are dealing with true Diracs.

Let us now define the interaction energy W . Let m > 0 be a given positive number(corresponding to the density of points). We say a vector field E belongs to the class Am if

(14)

E = ∇H −∆H = 2π(ν −m) for some ν =∑p∈Λ

δp, where Λ is a discrete set in R2.

As said above, the vector-field E physically corresponds, in the electrostatic analogy, to theelectric field generated by the point charges, and −E⊥ to a superconducting current in thesuperconductivity context.

Note that H has a logarithmic singularity near each a, and thus |∇H|2 is not integrable;however, when removing small balls of radius η around each a, adding back π log η, and lettingη → 0, this singularity can be “resolved”.

Definition 1. We define the renormalized energy W for E ∈ Am by

(15) W (E) := lim supR→∞

W (E,χBR)

|BR|,

where χBR is any cutoff function supported in BR with χBR = 1 in BR−1 and |∇χBR | ≤ C,and W (E,χ) is defined by

(16) W (E,χ) := limη→0

∫R2\∪ni=1B(xi,η)

χ|E|2 + π(log η)∑i

χ(xi).

The name is given by analogy with the “renormalized energy” introduced in [BBH] as theeffective interaction energy of a finite number of point vortices. Renormalized refers here tothe way the energy is computed by substracting off the infinite contribution corresponding tothe self interaction of each charge or vortex.

8

In the particular case where the configuration of points Λ has some periodicity, i.e. if itcan be seen as n points a1, · · · , an living on a torus T of appropriate size, then W can beexpressed much more simply as a function of the points only:

(17) W (a1, · · · , an) =π

|T|∑j 6=k

G(aj − ak) + π limx→0

(G(x) + log |x|) ,

where G is the Green’s function of the torus (i.e. solving −∆G = δ0 − 1/|T|). The Greenfunction of the torus can itself be expressed explicitely in terms of some Eisenstein series andthe Dedekind Eta function. The definition (15) thus allows to generalize such a formula toany infinite system, without any periodicity assumption.

The question of central interest to us is that of understanding the minimum and minimizersof W . Here are a few remarks.

1. The value of W doesn’t really depend on the cutoff functions satisfying the assumption.

2. W is unchanged by a compact perturbation of the points.

3. One can reduce by scaling to studying W over the class A1.

4. It can be proven that minimizers of W over A1 exist (and the minimum is finite).

5. It can be proven that the minimum of W is equal to the limit as N →∞ of the minimumof W over configurations of points which are N ×N periodic.

We do not know the value of minA1 W , however we can identify the minimum of W overa restricted class: that of points on a perfect lattice (of volume 1).

Theorem 3 ([SS5]). The minimum of W over perfect lattice configurations (of density 1) isachieved uniquely, modulo rotations, by the triangular lattice.

By triangular lattice, we mean the lattice Z + Zeiπ/3, properly scaled.The proof of this theorem uses the explicit formula for W in the periodic case in terms of

Eisenstein series mentioned above. By transformations using modular functions or by directcomputations, minimizing W becomes equivalent to minimizing the Epstein zeta functionζ(s) =

∑p∈Λ

1|p|s , s > 2, over lattices. Results from number theory in the 60’s to 80’s (due to

Cassels, Rankin, Ennola, Diananda, Montgomery, cf. [Mo] and references therein) say thatthis is minimized by the triangular lattice.

In view of the experiments showing Abrikosov lattices in superconductors, it is then nat-ural to formulate the

Conjecture 1. The “Abrikosov” triangular lattice is a global minimizer of W .

Observe that this question belongs to the more general family of crystallization problems.A typical such question is, given a potential V in any dimension, to determine the pointpositions that minimize ∑

i 6=jV (xi − xj)

(with some kind of boundary condition), or rather

limR→∞

1

|BR|∑

i 6=j,xi,xj∈BR

V (xi − xj),

9

and to determine whether the minimizing configurations are perfect lattices. Such questionsare fundamental in order to understand the crystalline structure of matter. They also arisein the arrangement of Fekete points, “Smale’s problem” on the sphere, or the “Cohn-Kumarconjecture”... One should immediately stress that there are very few positive results in thatdirection in the literature (in fact it is very rare to have a proof that the solution to some min-imization problem is periodic). Some exceptions include the two-dimensional sphere packingproblem, for which Radin [Ra] showed that the minimizer is the triangular lattice, and anextension of this by Theil [Th] for a class of very short range Lennard-Jones potentials. Herethe corresponding potential is rather logarithmic, hence long-range, and these techniques donot apply. The question could also be rephrased as that of finding

min “‖∑i

δxi − 1‖(H1)∗”

where the quantity is put between brackets to recall that δxi does not really belong to thedual of the Sobolev space H1 but rather has to be computed in the renormalized way thatdefines W . A closely related problem: to find

min ‖∑i

δxi − 1‖Lip∗

turns out to be much easier. It is shown in [BPT] with a relatively short proof that again thetriangular lattice achieves the minimum.

With S. Rota Nodari, in [RNS], we investigated further the structure of minimizers of W(or rather, a suitable modification of it) and we were able to prove that the energy densityand the points were uniformly distributed at any scale 1, in good agreement with (but ofcourse much weaker than!) the conjecture of periodicity of the minimizers.

Even though the minimization of W is only conjectural, it is natural to view it as (orexpect it to be) a quantitative “measure of disorder” of a configuration of points in the plane.In this spirit, in [BS] we use W to quantify and compute explicitly the disorder of some classicrandom point configurations in the plane and on the real line.

1.4.2 Next order result for GLε

We can now state the main next order result on GLε.

Theorem 4 ([SS5]). Consider minimizers (ψε, Aε) of the Ginzburg-Landau in the regimeλΩ|log ε| ≤ hex 1

ε2. After blow-up at scale

√hex around a randomly chosen point in ωλ,

the ∇hε converge as ε → 0 to vector fields in the plane which, almost surely, minimize W .Moreover, minGLε can be computed up to o(hex) (i.e. up to an error o(1) per vortex):

minGLε = hex2 min EMF

λ + (1− 1

2λ)hex|ωλ|min

A1

W + o(hex) as ε→ 0.

Thus, our study reduces the problem to understanding the minimization of W . If the laststep of proving Conjecture 1 was accomplished, this would completely justify the emergence ofthe Abrikosov lattice in superconductors, for applied magnetic fields in the regime consideredhere, which is hex Hc2 . Note that the Abrikosov lattice is also expect to appear for hex

up to Hc2 , but the mathematical mechanism for it is then different (instead of a nonlinearproblem, one can reduce to a minimization in the Lowest Landau Level, see e.g. [AS]).

10

1.5 Other results

We also investigated the structure of solutions to (GL) which are not necessarily minimizing,in other words critical points of (1). Let us list here the main results:

• In [Se2] and [SS1, Chap 12], we prove the existence of branches of local minimizers of(1) (hence stable solutions) of similar type as the solutions in Theorem 2 which havearbitrary bounded numbers of vortices all of degree +1 and exist for wide ranges of theparameter hex, and the locations of the vortices in these solutions are also characterized.In other words, for a given hex, there may exist an infinite number of stable solutionswith vortices, indexed by the number of vortices. Only one specific value of the numberof vortices is optimal, depending on the value of hex, as in Theorem 2.

• Similarly, there also exist multiple branches of locally minimizing solutions of (GL)with (rather arbitrary) unbounded numbers of vortices. This is proven in [CS]. Thesevortices arrange themselves according to a uniform density over a set again determinedby an obstacle problem, and at the microscopic level, they tend to minimize W .

• If one considers a general solution to (GL) with not too large energy, then one can char-acterize the possible distributions of the vortices, depending on whether their numberis bounded or unbounded as ε → 0. The characterization says roughly that the totalforce (generated by the other vortices and by the external field) felt by a vortex has tobe zero. In particular it implies that if there is a large number of vortices converging toa certain regular density, that density must be constant on its support. This is provenin [SS3] and [SS1, Chapter 13].

The analysis of the three-dimensional version of the Ginzburg-Landau model is of coursemore delicate than that of the two-dimensional one, because of the more complicated geometryof the vortices, which are lines instead of points (the first result attacking this question was[Ri]). This explains why it has taken more time for analogous results to be proven. The bestto date is the three-dimensional equivalent of Theorem 1 by Baldo-Jerrard-Orlandi-Soner[BJOS1, BJOS2], see also references therein and [Ka].

2 The 2D Coulomb gas

The connection with the jellium is what prompted us to examine in [SS6] the consequencesthat our study could have for the 2D classical Coulomb gas. More precisely, let us consider a2D Coulomb gas of n particles xi ∈ R2 in a confining potential V (growing sufficiently fast atinfinity), and let us take the mean-field scaling of interaction where the Hamiltonian is givenby

(18) Hn(x1, · · · , xn) = −∑i 6=j

log |xi − xj |+ n

n∑i=1

V (xi).

Note that ground states of this energy are also called “weighted Fekete sets”, they arise ininterpolation, cf. [ST], and are interesting in their own right.

For V (x) = |x|2, some numerical simulations, see Fig. 3, give the shapes of minimizers ofHn, which is then also a particular case of wn that appeared in (12).

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-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

Figure 3: Numerical minimization of Hn by Gueron-Shafrir [GS], n = 29

The Gibbs measure for the same mean-field Coulomb gas at temperature 1/β is

(19) dPβn(x1, · · · , xn) =1

Zβne−βHn(x1,··· ,xn)dx1 · · · dxn

where Zβn is the associated partition function, i.e. a normalization factor that makes dPβn aprobability measure. The particular case of β = 2 and V (x) = |x|2 corresponds to the lawof the eigenvalues for random matrices with iid normal entries, the Ginibre ensemble. Theconnection between Coulomb gases and random matrices was first pointed out by Wigner[Wi1] and Dyson [Dy]. For general background and references, we refer to [Fo]. The samesituation but with xi belonging to the real line instead of the plane is also of importance forrandom matrices, the corresponding law is that of what is often called a “β-ensemble”.

Among interacting particle systems, Coulomb gases have always been considered as par-ticularly interesting but delicate, due to the long range nature of the interactions (which isparticularly true in 1 and 2 dimensions). The case of one-dimensional Coulomb gases canbe solved more explicitly [Le, Ku, BL, AM], and crystallisation at zero temperature is es-tablished. In dimension 2, many studies rely on a rather “algebraic approach” with exactcomputations (e.g. [Ja]), or require a finite system or a condition ensuring local charge bal-ance [AJ, SM]. Our approach is strictly energy-based and this way valid for any temperatureand general potentials V .

2.1 Analysis of ground states

A rather simple analysis of (18), analogous to (7), leads to the result that minimizers of Hn

satisfy∑ni=1 δxin → µ0 where µ0 minimizes the following mean-field limit for Hn/n

2 as n→∞:

(20) F(µ) =

∫R2×R2

− log |x− y| dµ(x) dµ(y) +

∫R2

V (x) dµ(x)

defined for µ a probability measure. The unique mean-field minimizer, which is also called theequilibrium measure in potential theory is a probability µ0 (just as the minimizer of EMF

λ for

12

Ginzburg-Landau, it can also be viewed as the solution of an obstacle problem). Its support,that we will denote Σ, is compact (and assumed to have a nice boundary). For example ifV (x) = |x|2, it is a multiple of the characteristic function of a ball (the circle law for theGinibre ensemble in the context of random matrices), and this is analogous to Theorem 1 andthe obstacle problem distribution found for Ginzburg-Landau. Deriving this mean-field limitis significantly easier than for Ginzburg-Landau, due to the discrete nature of the startingenergy, and the fact that all charges are +1 (as opposed to the vortex degrees, which can beany integer).

The connection with the Ginzburg-Landau situation is made by defining analogously thepotential generated by the charge configuration using the mean-field density µ0 as a neu-tralizing background, this yields the following equation playing the role of the analogue to(3):

hn = −2π∆−1( n∑i=1

δxi − nµ0

)in R2.

The next step is again to express this in the blown-up coordinates at scale√n (analogous to

the√hex scale previously) around x0, x′ =

√n(x− x0), via h′n the solution to

(21) h′n(x′) = −2π∆−1( n∑i=1

δx′i − µ0(x0 +x′√n

)).

When taking n→∞, the limit equation to (21) is

(22) −∆h = 2π(∑

a

δa − µ0(x0))

in R2

analogue of (13), corresponding to another infinite jellium with uniform neutralizing back-ground µ0(x0).

Expanding the energy to next order is done via a suitable splitting, by analogy withGinzburg-Landau. In fact in this setting the splitting procedure is quite simple: it suffices towrite νn :=

∑ni=1 δxi as nµ0 + (νn − nµ0). Noting that

Hn(x1, · · · , xn) =

∫∫4c− log |x− y| dνn(x) dνn(y) +

∫V (x) dνn(x)

where 4 denotes the diagonal, inserting the indicated splitting of νn, we eventually find theexact decomposition

(23) Hn(x1, · · · , xn) = n2F(µ0)− n

2log n+

1

πW (∇h′n,1R2) + 2n

n∑i=1

ζ(xi),

where W (E,χ) is as in (16). The function ζ in (23) is explicit and determined only by V , it islike an effective potential and plays no other role than confining the particles to Σ = Supp(µ0)(it is zero there, and positive elsewhere), so there remains to understand the limit n → ∞of W (∇h′n,1R2) corresponding to (22). One of our main results below is that this term is oforder n. An important advantage of this formulation is that it transforms, via (23), the sumof pairwise interactions into an extensive quantity in space (16), which allows for localizing(via a screening procedure), cutting and pasting, etc...

Let us now state the next order result playing the role of Theorem 4.

13

Theorem 5 ([SS6]). Let (x1, . . . , xn) ∈ (R2)n.Up to extraction of a subsequence, we have thatPn, the push-forward of 1

|Σ|dx|Σ (the normalized Lebesgue measure on Σ) by

x 7→(x,En(

√nx+ ·)

), En := ∇h′n

converges weakly in the sense of measures to some probability measure P whose first marginalis 1|Σ|dx|Σ, and satisfying that P -a.e. (x,E) ∈ Aµ0(x). In addition, along this subsequence, we

have

(24) lim infn→∞

1

n

(Hn(x1, . . . , xn)− n2F(µ0) +

n

2log n

)≥ |Σ|

π

∫W (E) dP (x,E).

Moreover, this lower bound is sharp, and for minimizers of Hn, it holds that P -a.e. (x,E),E minimizes W over Aµ0(x) and

(25) limn→∞

1

n

(minHn − n2F(µ0) +

n

2log n

)=|Σ|π

∫ (min

E∈Aµ0(x)

W)dP (x,E)

=1

π

∫Σ

minAµ0(x)

W := α0.

This result contains a sharp lower bound valid for any configuration, and not just for mini-mizers. The lower bound is by W (P ) := |Σ|

π

∫W (E) dP (x,E), an average of the renormalized

energy W with respect to all the possible blow-up centers, and we notice that α0 = min W (P )where the minimum is taken over the P ’s whose first marginal is 1

|Σ|dx|Σ and which satisfy

that P -a.e. (x,E) ∈ Aµ0(x). A rephrasing is that minimizers of Hn provide configurations ofpoints in the plane whose associated “electric fields” E minimize, after blow-up and taking thelimit n → ∞, the renormalized energy, P−a.e., i.e. (heuristically) for almost every blow-up

center. W is a next-order limiting energy for Hn (or next order Γ-limit, in the language ofΓ-convergence). It is the term that sees the difference between different microscopic patternsof points, beyond the macroscopic averaged behavior µ0.

If again the conjecture on the minimizers of W was established, this would prove thatpoints in zero temperature Coulomb gases should form a crystal in the shape of an Abrikosovtriangular lattice.

The result of Theorem 5 can be improved when one looks directly at minimizers (orground states) of Hn instead of general configurations. With S. Rota Nodari, we obtainedthe following

Theorem 6 ([RNS]). Let (x1, . . . , xn) be a minimizer of Hn. Let x′i, ∇h′n, Σ, µ0, be as aboveand Σ′ =

√nΣ, µ′0(x) = µ0(x/

√n). The following holds, with K`(a) denoting the square of

sidelength ` centered at a:

1. for all i ∈ [1, n], xi ∈ Σ;

2. there exist β ∈ (0, 1), c > 0, C > 0 (depending only on µ0), such that for every ` ≥ cand a ∈ Σ′ such that d(K`(a), ∂Σ′) ≥ nβ/2, we have

(26) lim supn→∞

1

`2

∣∣∣∣∣W (∇h′n, χK`(a))−∫K`(a)

(minAµ′0(x)

W)dx

∣∣∣∣∣ ≤ o(1) as `→ +∞,

14

where χK`(a)is any cutoff function supported in K`(a) and equal to 1 in K`−1(a); and

(27) lim supn→∞

∣∣∣∣∣#(K`(a) ∩ x′i)−∫K`(a)

m′0(x) dx

∣∣∣∣∣ ≤ C`.This says that for minimizers, the configurations seen after blow-up around any point

sufficiently inside Σ (and not just a.e. point) tend to minimize W and their points follow thedistribution µ0 even at the microscopic scale.

This result is to be compared with previous results of Ameur - Ortega-Cerda [AOC] where,using a very different method based on “Beurling-Landau densities”, they prove (27), witha larger possible error o(`2) but for distances to ∂Σ′ which can be smaller (their paper doesnot however contain the connection with W ).

When (18) is considered for xi ∈ R instead of R2, then it is the Hamiltonian of whatis called a “log gas”. The same corresponding result are proven in [SS7], together with thedefinition of an appropriate one-dimensional version of W , for which the minimum is thistime shown to be achieved by the lattice configuration (or “clock distribution”) Z.

2.2 Method of the proofs

The proof of the above Theorems 4 and 5 follows the idea of Γ-convergence (see e.g. [Br, DM])i.e. relies on obtaining general (i.e. ansatz-free) lower bounds, together with matching upperbounds obtained via explicit constructions.

There are really two scales in our energies: a macroscopic scale (that of the support of µ∗or µ0), and the scale of the distance between the points (or vortices) which is much smaller.We know how to obtain lower bounds for the energy at the microscopic scale, but it is notclear in our case how to “glue” these estimates together. For that purpose we introduced in[SS5] a new general method for obtaining lower bounds on two-scale energies. A probabilitymeasure approach allows to integrate the local estimates via the use of the ergodic theorem(an idea suggested by S. R. S. Varadhan). That abstract method applies well to positive (orbounded below) energy densities, but those associated to W (E,χ) are not, due to one of themain difficulties mentioned above: the lack of local charge neutrality. To remedy this we startby modifying the energy density to make it bounded below, using sharp energy lower boundsby improved “ball construction” methods (a la Jerrard / Sandier).

The method is the same for both cases but in the case of the Coulomb gas, it is complicatedby the (slow) variation of the macroscopic density µ0. For Ginzburg-Landau, the situationis on the other hand made more difficult by the presence of vortices of arbitrary signs anddegrees.

Finally, let us mention that in [GMS1, GMS2] we carry out a similar analysis for the“Ohta-Kawasaki” model of diblock-copolymers, where the interacting objects are this time“droplets” that can have more complicated geometries and nonquantized charges (their chargeis really their volume), and derive the same next order limiting energy W .

2.3 The case with temperature

Understanding the asymptotics of Hn via Theorem 5 (and not only of ground states) naturallyallows to deduce information on finite temperature states. First, inserting the lower bound onHn found in Theorem 5 into (19) directly yields an upper bound on Zβn . Conversely, using an

15

explicitly constructed test-configuration meant to approximate minimizers of Hn up to o(n),and showing that a similar upper bound holds in a sufficiently large phase-space neighborhoodof that configuration, allows to deduce a lower bound for Zβn . The lower bound and the upperbound will coincide as β →∞ only. The main statement is

Theorem 7 ([SS6]). For any β > 0 there exists Cβ such that limβ→+∞Cβ = 0 and

lim supn→∞

1

nβ

∣∣∣∣logZβn +nβ

2

(nF(µ0)− 1

2log n+ α0

)∣∣∣∣ ≤ Cβ,where

α0 =1

π

∫minAµ0(x)

W dx.

Only the term in n2 of this expansion was previously known, for such a general situationof general β and V . This is in contrast to the one-dimensional log gas case where Zβn is knownfor V quadratic and all β by Selberg integrals, and for more general V ’s as well. Also, theresult relates the computation of Zβn to that of the unknown constant minW , so to proveConjecture 1 it would suffice in principle to know how to compute Zβn for a 2D Coulomb gas!

The final result is a large deviations type result. First, let us recall the best previouslyknown result which is a result of “large deviations from the circle law”:

Theorem 8 (Petz-Hiai [PH], Ben Arous-Zeitouni [BZ], ). Pβn satisfies a large deviationsprinciple with good rate function F(·) and speed n−2: for all A ⊂ probability measures,

− infµ∈A

F(µ) ≤ lim infn→∞

1

n2logPβn(A) ≤ lim sup

n→∞

1

n2logPβn(A) ≤ − inf

µ∈AF(µ),

where F = F −minF .

This thus says that the probability of an event A is exponentially small if F > minF inA, i.e. if one deviates from the circle law µ0:

Pβn(A) ≤ e−n2 infA(F−F(µ0)).

One may wonder whether the same is true at next order, i.e. whether the arrangementsof points after blow up follow the next order optimum of Hn, i.e. minimize W . Figure 4,corresponding to the Ginibre case of β = 2 and V (x) = |x|2 indicates that this should not bethe case since the points do not arrange themselves according to triangular lattices.

We may then wonder how to quantify the order or rigidity of the configurations accordingto the temperature. The following result provides some answer, and an improvement at nextorder on Theorem 8:

Theorem 9 ([SS6]). For any β > 0 there exists Cβ > 0 such that limβ→+∞Cβ = 0 and thefollowing holds. For An ⊂ (R2)n, we have

lim supn→∞

1

nlogPβn(An) ≤ −β

( |Σ|π

infP∈A

∫W (E)dP (x,E)− α0 − Cβ

),

and A is the set of probability measures which are limits (in the weak sense) of blow-ups atrate

√n around a point x of the electric fields ∇hn associated to

∑ni=1 δxi with (xi) ∈ An.

16

Figure 4: Eigenvalues of 1000-by-1000 matrix with i.i.d Gaussian entries (β = 2)(from Benedek Valko’s webpage)

Modulo again the conjecture on minW , this proves crystallization as the temperature goesto 0: after blowing up around a point x in the support of µ0, at the scale of

√n, as β →∞ we

see (almost surely) a configuration which minimizes W . Indeed, α0 is the minimum value that

W can possibly take, and is achieved if and only if W (E) = minAµ0(x)W for P -a.e. (x,E).

For β finite, the result says that the average of W lies below a fixed threshhold (sayα0 + 1 + C

β ), except with very small probability.To our knowledge, this is the first time Coulomb gases are rigorously connected to triangu-

lar lattices (again modulo the solution to the conjecture on the minimum of W ), in agreementwith predictions in the literature (cf. [AJ] and references therein).

3 Higher dimensional Coulomb Gases

With Nicolas Rougerie, in [RS], we extended the results for the Coulomb gas presented aboveto arbitrary higher dimension, considering this time

Hn(x1, . . . , xn) =∑i 6=j

g(xi − xj) + nn∑i=1

V (xi)

with xi ∈ Rd and the kernel g is the Coulomb kernel − log |x| in dimension 2 and |x|2−d indimension d ≥ 3. The mean-field limit energy is defined in the same way by

F(µ) =

∫∫Rd×Rd

g(x− y) dµ(x) dµ(y) +

∫RdV (x) dµ(x).

Turning to higher dimension required a new approach and a new definition of W , theprevious one being very tied to the two-dimensional “ball construction method” as alluded toin Section 2.2. The new approach is based on a different way of renormalizing, or substractingoff the infinite “self-interaction” energy of each point: we replace point charges by smeared-out charges, as in “Onsager’s lemma”. More precisely, we pick some arbitrary fixed radial

17

nonnegative function ρ, supported in B(0, 1) and with integral 1, and for any point x andη > 0 we introduce the smeared charge

δ(η)x =

1

ηdρ

(·η

)∗ δx.

Newton’s theorem asserts that the Coulomb potentials generated by the smeared charge δ(η)x

and the point charge δx coincide outside of B(x, η). A consequence of this is that if we defineinstead of

(28) h′n(x′) = −cd∆−1

(n∑i=1

δx′i − µ0(x0 + x′n−1/d)

)

as in (21), the potential

(29) h′n,η(x′) = −cd∆−1

(n∑i=1

δ(η)x′i− µ0(x0 + x′n−1/d)

)

then h′n and h′n,η coincide outside of the B(x′i, η). Moreover, they differ by∑

i fη(x − x′i)where fη is a fixed function equal to cd∆

−1(δ(η)0 − δ0), vanishing outside B(0, η). (Here the

constant cd is the constant such that −∆g = cdδ0, depending only on dimension).By keeping these smeared out charges, we are led at the limit to solutions to

(30) −∆hη = cd

(∑a

δ(η)a −m

), m = µ0(x0)

which are in bijection with the functions h solving the same equation with η = 0, via addingor subtracting

∑a fη(x− a).

For any fixed η > 0 one may then define the electrostatic energy per unit volume of theinfinite jellium with smeared charges as

(31) lim supR→∞

−∫KR

|∇hη|2 := lim supR→∞

|KR|−1

∫KR

|∇hη|2

where hη is as in the above definition and KR denotes the cube [−R,R]d. This energy is nowwell-defined for η > 0 and blows up as η → 0, since it includes the self-energy of each smearedcharge in the collection. One should then “renormalize” (31) by removing the self-energy ofeach smeared charge before taking the limit η → 0. The leading order energy of a smearedcharge is κdg(η) where κd is a constant depending on dimension and on the choice of thesmoothing function ρ. We are then led to the definition

Definition 2 ([RS]). The renormalized energy W is defined over the class Am by

W(∇h) = lim infη→0

Wη(∇h) = lim infη→0

(lim supR→∞

−∫KR

|∇hη|2 −m(κdg(η) + γ21d=2)

),

where κd and γ2 are constants depending only on the choice of ρ.

18

It is also proven in [RS] that W achieves its minimum (for each given m), which indimension 2 coincides with that of W . It is also natural to expect that the minimum of Wmay be achieved by crystalline configurations (like the FCC lattice in three dimensions) butthis is a completely open question.

We are able to show that a similar splitting relation as (23) holds, which makes appearWη instead of W . It is however only an inequality, but equality is retrieved as one lets η → 0.This allows to let n → ∞ and obtain lower bounds via the same “probabilistic method”mentioned in Section 2.2, for fixed η. At the end we let η tend to 0, to retrieve similar resultsas Theorems 7-9.

The following gives the analogue to Theorem 7 but this time expressed in terms of thefree energy Fn,β = − 2

β logZβn .

Theorem 10 ([RS]).Let us define

(32) ξd :=

1

cd

(minA1

W)∫

Rdµ

2−2/d0 if d ≥ 3

1

2πminA1

W − 1

2

∫R2

µ0 logµ0 if d = 2.

Let β = lim supn→+∞ βn1−2/d and assume β > 0. There exists Cβ with limβ→+∞Cβ = 0

such that, for n large enough,

(33)∣∣∣Fn,β − n2F(µ0) +

(n2

log n)1d=2 − n2−2/dξd

∣∣∣ ≤ Cβn2−2/d.

Taking in particular formally β = ∞ leads to Fn,∞ = minHn and allows to retrieve thenext order expansion of minHn.

An analogue of Theorem 9 is also given:

Theorem 11 ([RS]). Let β > 0 be as in Theorem 10. There exists Cβ with limβ→+∞Cβ = 0

such that, for any An ⊂ (Rd)n, it holds that

(34) lim supn→∞

logPβn(An)

n2−2/d≤ −β

2

(|Σ|cd

infP∈A∞

∫W(E)dP (x,E)− ξd − Cβ

),

where A is the set of probability measures which are limits (in the weak sense) of ∇h′n(x+ ·)associated to the (xi) ∈ An via (28).

These results show that the expected regime for crystallization (maybe surprisingly) de-pends on the dimension and is the regime β n2/d−1. In that regime, the Gibbs measureessentially concentrates on minimizers of W, which as before would show crystallization ifone knew that such minimizers have to be crystalline.

For a self-contained presentation of these topics, one can also refer to [Se4].

Other than proving that specific crystalline configurations achieve the minimum of Wor W, the main result missing in these studies is to establish a complete Large DeviationsPrinciple at next order for the Coulomb gas, and identifying the right rate function whichshould involve W , but not only. This would prove at the same time the existence of a completenext order “thermodynamic limit” (for leading order results see [LN] and references therein.

Let us finish by pointing out that the quantum case (of the Coulomb gas) is quite differentand studied in [LNSS]: the next order term is of order n and identified to be the ground stateof the “Bogoliubov Hamiltonian.”

19

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S. SerfatyUPMC Univ Paris 06, UMR 7598 Laboratoire Jacques-Louis Lions,Paris, F-75005 France ;CNRS, UMR 7598 LJLL, Paris, F-75005 France& Courant Institute, New York University251 Mercer st, NY NY 10012, USA.E-mail: serfaty@ann.jussieu.fr

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